Let n, N ∈ N with Ω ⊆ R n open. Given H ∈ C 2 (Ω × R N × R N n ), we consider the functional
The associated PDE system which plays the role of EulerHerein we establish that generalised solutions to (2) can be characterised as local minimisers of (1) for appropriate classes of affine variations of the energy. Generalised solutions to (2) are understood as D-solutions, a general framework recently introduced by one of the authors.

In this paper we consider the PDE system of vanishing normal projection of the Laplacian for C 2 maps u :This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the p-Laplace system for all p ∈ [2, ∞]. For p = ∞, the ∞-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial calculus of variations in L ∞ . Herein we show that the image of a solution u is piecewise affine if either the rank of Du is equal to one or n = 2 and u has additively separated form. As a consequence we obtain corresponding flatness results for p-Harmonic maps for p ∈ [2, ∞].

We discuss two distinct minimality principles for general supremal first order functionals for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson's standard notion of absolute minimisers and the concept of ∞-minimal maps introduced more recently by the second author. We prove that C 1 absolute minimisers characterise a divergence system with parameters probability measures and that C 2 ∞-minimal maps characterise Aronsson's PDE system. Since in the scalar case these different variational concepts coincide, it follows that the non-divergence Aronsson's equation has an equivalent divergence counterpart.

We derive Onsager–Machlup functionals for countable product measures on weighted ℓ
p
subspaces of the sequence space
R
N
. Each measure in the product is a shifted and scaled copy of a reference probability measure on
R
that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

We discuss two distinct minimality principles for general supremal first order functionals for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson's standard notion of absolute minimisers and the concept of ∞-minimal maps introduced more recently by the second author. We prove that C 1 absolute minimisers characterise a divergence system with parameters probability measures and that C 2 ∞-minimal maps characterise Aronsson's PDE system. Since in the scalar case these different variational concepts coincide, it follows that the non-divergence Aronsson's equation has an equivalent divergence counterpart.

In this paper we consider the PDE system of vanishing normal projection of the Laplacian for C 2 maps u :This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the p-Laplace system for all p ∈ [2, ∞]. For p = ∞, the ∞-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial Calculus of Variations in L ∞ . Herein we show that the image of a solution u is piecewise affine if either the rank of Du is equal to one or n = 2 and u has the additively separated form u(x, y) = f (x) + g(y). As a consequence we obtain corresponding flatness results for the images of p-Harmonic maps, p ∈ [2, ∞].

The aim of this work is to derive new explicit solutions to the ∞-Laplace equation, the fundamental PDE arising in Calculus of Variations in the space L ∞ . These solutions obey certain symmetry conditions and are derived in arbitrary dimensions, containing as particular sub-cases the already known classes of two-dimensional infinity-harmonic functions.

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